Quantum simulation of hyperbolic space with circuit quantum electrodynamics: From graphs to geometry

Hyperbolic band theory

“…Using graph theory and numerical diagonalization, Kollár et al (10) obtained general mathematical results concerning the existence of extended degeneracies and gaps in the spectrum of tight-binding Hamiltonians on a variety of discrete hyperbolic lattices. Boettcher et al (11) developed a hyperbolic analog of the effective-mass approximation in solid-state physics, showing that such tight-binding Hamiltonians reduce in the long-distance limit to the hyperbolic Laplacian-the Laplace-Beltrami operator associated with the Poincaré metric on the hyperbolic plane-and proposing the synthetic structures of Kollár et al (7) as a new platform for the simulation of quantum field theory in curved space. Topological quantum phenomena in hyperbolic lattices were explored using real-space numerical diagonalization by Yu et al (12).…”

Gaplessness of Landau Hamiltonians on Hyperbolic Half-planes via Coarse Geometry

“…However, it has remained an open question, even for the paradigmatic Landau Hamiltonian, whether the gap-filling phenomenon occurs in hyperbolic (or more general) geometries. With the ability to effectively simulate dynamics in hyperbolic geometry [3,15], this has become a pertinent question to address, and our Theorem 1 answers this in the affirmative.…”

“…with respect to the even-odd grading of the spinor bundle. Here the index of / D + turns out to be a topological invariant, which is calculated by the celebrated Atiyah-Singer index theorem 3 . While / D + is unbounded, the operator…”

“…similar to the finite-dimensional case, where ker( / D ± a ) ∈ C * (X ) denotes the orthogonal projection onto ker( / D ± a ). 3 Since / D is self-adjoint, the index of / D − satisfies Ind( / D − ) = −Ind( / D + ) and gives nothing new.…”

Quantum walks simulating non-commutative geometry in the Landau problem

“…24-27 and the symplectic discrete geometry behind some QWs has been presented in Ref. 26. The interface between QWs and geometry has also been explored on QWs defined on various structures with interesting geometries, be it different types of lattices [28][29][30][31][32] or Cayley graphs, [33][34][35][36] including Cayley graphs of non-Abelian groups. In several cases, a continuum limit exists and coincides with physics or mathematics on known differential manifolds.…”